SIMSON. 507 



by Dr. Traill, when studying under the Professor. 

 There may thus be said to be in all ninety-eight pro- 

 positions. The four lemmas are propositions ancillary 

 to the author's own investigations ; for many of his 

 theorems are the lemmas preserved by Pappus as an- 

 cillary to the porisms of Euclid. 



In all these investigations the strictness of the 

 Greek geometry is preserved almost to an excess ; and 

 there cannot well be given a more remarkable illus- 

 tration of its extreme rigour than the very outset of 

 this great work presents. The porism is, that a point 

 may be found in any given circle through which all 

 the lines drawn cutting its circumference and meeting 

 a given straight line shall have their segments within 

 and without the circle in the same ratio. This, 

 though a beautiful proposition, is one very easily 

 demonstrated, and is, indeed, a corollary to some of 

 those in the ' Elements.' But Dr. Simson prefixes a 

 lemma : that the line drawn to the right angle of a 

 triangle from the middle point of the hypotenuse, is 

 equal to half that hypotenuse. Now this follows, 

 if the segment containing the right angle be a semi- 

 circle, and it might be thought that this should be 

 assumed only as a manifest corollary from the pro- 

 position, or as the plain converse of the proposition, 

 that the angle in a semicircle is a right angle, but 

 rather as identical with that proposition ; for if we 

 say the semicircle is a right-angled segment, we also 

 say that the right-angled segment is a semicircle. 

 But then it might be supposed that two semicircles 

 could stand on one base : or, which is the same thing, 

 that two perpendiculars could be drawn from one 



